www.ietdl.org Published in IET Microwaves, Antennas & Propagation Received on 15th December 2010 doi: 10.1049/iet-map.2010.0607
ISSN 1751-8725
Constrained parameter extraction for microwave design optimisation using implicit space mapping S. Koziel1 J.W. Bandler 2 Q.S. Cheng 2 1
Engineering Optimization & Modeling Center, School of Science and Engineering, Reykjavik University, Menntavegur 1, 101 Reykjavik, Iceland 2 Simulation Optimization Systems Research Laboratory, Department of Electrical and Computer Engineering, McMaster University, Hamilton, ON, Canada L8S 4K1 E-mail:
[email protected]
Abstract: To date, space mapping remains one of the most efficient design optimisation methodologies in microwave engineering. Still, its performance depends on the underlying surrogate model, in particular, its approximation and generalisation capabilities. By proper selection of the space mapping transformations and their parameters, a trade-off between these can be obtained. Often, this is done by trial and error that may lead to excessive computational overhead and poor quality of the optimisation outcome. In this study, an adaptively constrained parameter extraction is introduced. Based on convergence results for space mapping algorithms, it allows us to automatically find the approximation-generalisation trade-off through the adjustment of the surrogate model parameter space. Improved performance of the space mapping algorithm is obtained both in terms of convergence properties and the quality of the optimised design. Algorithm convergence is additionally improved by constraining the surrogate optimisation process. The authors’ technique is validated using several microwave design problems.
1
Introduction
Simulation-based optimisation and design closure play an increasing role in contemporary microwave engineering. This is because theoretical models typically provide only initial designs, which need further tuning to meet given performance specifications. For some classes of microwave structure (e.g. substrate-integrated circuits, ultrawide-band antennas) there are no systematic design procedures to realise designs satisfying given specification requirements. Here, EM-simulation-based design is the only feasible approach. Efficient simulation-driven design can be realised using the surrogate-based optimisation principle [1, 2], where the optimisation burden is shifted to a surrogate model, a computationally cheap representation of the structure being optimised (the ‘fine’ model). Successful surrogate-based approaches used in microwave area include space mapping [3 – 18], various forms of tuning [19 – 23] and tuning space mapping [24 –26], as well as response correction methods [27, 28]. The main idea behind space mapping optimisation is to shift the optimisation burden from an expensive ‘fine’ model to a cheap ‘coarse’ model by iteratively optimising and updating a surrogate built from the coarse model and available fine model data. If the coarse model represents the fine model sufficiently accurately, space mapping optimisation may yield a satisfactory solution after only a few fine model evaluations. For good performance of the space mapping algorithm, the surrogate model must have a good approximation capability 1156 & The Institution of Engineering and Technology 2011
(so that the surrogate can be aligned with the fine model) and, at the same time, a good generalisation capability (so that it is sufficiently accurate outside the training set) [29]. Unfortunately, finding the right combination of space mapping transformations and their parameters for a given design problem is not trivial [29], although the selection process can be automated to some extent [30, 31]. On the other hand, convergence properties of the algorithm can be improved by using trust region approach [32] which, however, increases the computational cost of the optimisation process. A novel approach to control the approximationgeneralisation trade-off of the surrogate model was proposed in [33]. It exploited an over-flexible surrogate model with very good approximation capability and whose generalisation properties were adjusted by reducing the parameter space of the surrogate. As a result, improved performance of the space mapping algorithm was obtained both with respect to its convergence properties and the quality of the optimised design. The basic implementation of the adaptively constrained parameter extraction (PE) of [33] is expanded here to explicitly control convergence of the space mapping algorithm by imposing constraints on surrogate model optimisation. We also generalise it to accommodate other space mapping transformations into the implicit space mapping framework. We present a theoretical derivation of our method which is based on general convergence results for space mapping algorithms [29]. Its relation to a trust-region approach is IET Microw. Antennas Propag., 2011, Vol. 5, Iss. 10, pp. 1156–1163 doi: 10.1049/iet-map.2010.0607
www.ietdl.org discussed. Robustness and computational efficiency are demonstrated using several microwave design problems. Comparisons with standard space mapping and with space mapping enhanced by a trust region approach are also given.
2
Implicit space mapping
Without loss of generality, we restrict our considerations to implicit space mapping [34], the most general approach to space mapping. Implicit space mapping allows us to introduce any number of surrogate model parameters. In particular, over-flexibility of the surrogate model required by our method can be easily obtained. Also, implicit space mapping can incorporate most other space mapping types, including input and output, as indicated in Section 2.2. 2.1
Implicit space mapping algorithm
The microwave design optimisation problem is defined as x∗f = arg min U (Rf (x)) x
(1)
where Rf: Xf R m, Xf # R n, denotes the response vector of a fine model of the device of interest; U: R m R is a given objective function, for example, minimax. Let Rc: Xc × Xp R m denote the response vector of the coarse model that describes the same object as the fine model: less accurate but much faster to evaluate. Rc depends on two sets of parameters: (i) design variables x, the same as in the fine model, and (ii) preassigned parameters xp , that is, parameters that are normally fixed in the fine model (e.g. dielectric constants) but can be adjusted in the coarse model to match it to Rf . We consider an optimisation algorithm that generates a sequence of designs x(i), i ¼ 1, 2, . . . , so that [34]
2.3
Surrogate model selection
The performance of space mapping algorithms heavily depends on the quality of the coarse model utilised in the optimisation process and the type of mapping involved in creating the surrogate model [29]. For implicit space mapping, the number of possible ways of selecting preassigned parameters is virtually unlimited. However, it is not obvious how to select a set of parameters that might allow the surrogate to closely approximate the fine model and simultaneously have good generalisation capability. A wrong choice of the parameter set may result in inadequate performance of the space mapping algorithm, including convergence issues, poor quality of the final design and excessive computational cost [30]. Various ways of alleviating this problem have been proposed. They include adaptive space mapping algorithms [30], and assessment methodologies [29, 31]. Although useful in the selection of the surrogate model and its parameters, none of these techniques guarantee algorithm convergence and overall good performance. On the other hand, trust-region-enhanced space mapping algorithms ensure algorithm convergence but not always a sufficient quality of the final design because in the case of inadequate improvement of the objective function [32], they force the algorithm to terminate. Also, the trust-region approach increases the computational cost of the space mapping optimisation process.
3
Constrained space mapping
The technique described in this section controls the trade-off between the approximation and generalisation capability of the surrogate model by properly restricting the parameter space of the model. The original version of the method was introduced in [33]. Here, we enhance it through constrained optimisation of the surrogate model. We also provide a theoretical background to our technique using rigorous convergence results for space mapping [29].
x(i+1) = arg min U (Rc (x, x(i) p ))
(2)
(i) (i) x(i) p = arg minRf (x ) − Rc (x , xp )
(3)
3.1 Convergence of the space mapping algorithm. Constrained space mapping concept
To obtain the best possible match with Rf at x(i), the values of the model parameters are updated in each iteration.
In [29], convergence results are formulated for the general space mapping algorithm (2), (3) assuming the following conditions (with Xf ¼ Xc ¼ X ). They can be reformulated (simplified) for the implicit space mapping case as follows.
x
where xp
2.2 Incorporating other space mapping transformations Although the algorithm presented here is formulated in terms of implicit space mapping, it can easily incorporate other popular space mapping types. For example, the surrogate model of the most popular input space mapping [3] is defined as Rs(x) ¼ Rc(x + c), where c is an n × 1 vector of parameters. It can be formulated in implicit-space-mapping terms as Rc.i (x, xp) where Rc.i is the implicit-spacemapping-like coarse model defined as Rc.i(x, xp) ¼ Rc(x + xp). In particular, the same software implementation can be used to realise our algorithm for both implicit and input cases. Analogous reformulations can be defined for scaling-like input space mapping Rs(x) ¼ Rc(B.x), output [4] and frequency space mapping [5]. Thus, the results of this paper are valid for all space mapping types. Some combinations of implicit and input space mappings are exploited in our examples (Section 4). IET Microw. Antennas Propag., 2011, Vol. 5, Iss. 10, pp. 1156– 1163 doi: 10.1049/iet-map.2010.0607
Assumption 1: Suppose that X and Xp are closed sets and that the following conditions are satisfied: 1. a set Xc∗ (xp) of solutions to the surrogate model optimisation problem xc∗ [ argmin {x [ X: U(Rc(x,xp)}, is not empty for any xp [ Xp and the following condition holds for any xp , yp [ Xp sup
sup x − y ≤ K(xp )xp − yp
x[Xc∗ (xp ) y[Xc∗ (yp )
(4)
where K: Xp R+ is a bounded function on Xp . 2. Let Xp∗ (x) denote a set of solutions to the surrogate model PE problem. We assume that Xp∗ (x) is not empty. We also assume that there is a k . 1 such that for each i . k, any ∗ (i) (i+1) x(i) [ Xp∗ (x(i+1)) there exist p [ Xp (x ) and any xp 1157
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www.ietdl.org Mi . 0 such that
select solutions that realise small Mi in K in (4) and (5), respectively.
(i+1) − x(i) − x(i) x(i+1) p p ≤ Mi x
(5) 3.2
where {x } is the sequence produced by algorithm (2), (3).
Initial surrogate model
(i)
Assumption 1 is discussed in detail in [29]. We only note here that condition (i) requires that the optimal solution of the surrogate model be regular with respect to the space mapping parameters; condition (ii) means that the surrogate model parameters at two subsequent iterations are sufficiently similar in value and this similarity is related to the distance between the subsequent iteration designs, that is, to x(i+1) – x(i). The following convergence results hold (the proof can be found in [29]). Theorem 1: Let {(x(i), x(i) p )} be a sequence defined by algorithm (2), (3). Suppose that Assumption 1 is satisfied and for each i . k with k as in Assumption 1 (ii) we have qi = K(x(i) p ) · Mi , 1 − 1
(6)
where 0 , 1 , 1 is a small constant independent of i. Then, the sequence {x(i)} is convergent to x∗ , where x∗ [ X, and the ∗ ∗ sequence {x(i) p } is convergent to xp , where xp [ Xp . Assumption 2: Suppose that Assumption 1, the set Xf∗ of solutions to (1) is not empty, and the following conditions are satisfied: 1. For any xp [ Xp there is x [ X such that U(Rc(x, xp)) ≤ Umin , where Umin ¼ min{x [ Xf: U(Rf(x))}. 2. Let (x∗ ,xp∗ ) be a limit point of the sequence {(x(i), x(i) p )} as in Theorem 1 and Rf(x∗ ) –Rc(x∗ , xp∗ ) ¼ 0 (i.e. the PE error is zero at the limit point). Corollary 1: Suppose that Assumption 2 is satisfied, and Rc , U are continuous. Then x∗ [ Xf∗ , that is, x∗ is an optimal solution of the fine model defined by (1). As follows from Theorem 1, convergence of the space mapping algorithm depends on the two fundamental inequalities (4) and (5) which describe, in a way, the generalisation capability of the surrogate model. The function K in (4) and constants Mi in (5) depend on both Rc and a specific set of space mapping parameters. On the other hand, convergence to a satisfactory design (i.e. the one that is the fine model optimum or at least something close to it) depends on the approximation capability of the surrogate model formulated in Assumption 2. Thus, in order to improve convergence and overall robustness of the space mapping algorithm one needs (i+1) to reduce the value of Mi in x(i+1) – x(i) – x(i), p p ≤ Mix reduce the value of K in (4) as well as ensure good approximation capability of the surrogate, that is, a small value of Rf(x(i))– Rc(x(i), x(i) p ) in each iteration. All of this can be achieved (to some extent) by using a surrogate model with a sufficient number of parameters, and by introducing (adaptive) control over both the PE and surrogate optimisation processes. A large number of parameters (here, preassigned parameters) will ensure good approximation but also some redundancy and nonuniqueness of the solutions to (2) and (3). At the same time, imposing constraints in both (2) and (3) allows us to 1158 & The Institution of Engineering and Technology 2011
It is assumed that the initial surrogate model, that is, the coarse model without any constraints on its preassigned parameters, is able to approximate the fine model with sufficient accuracy. This accuracy can be measured at a given iteration point x(i), using any suitable criteria, for example, 1(i) ¼ Rf(x(i)) – Rc(x(i), x(i) where .p p )p , determines the norm type (e.g. .2 for the Euclidean norm, or ./ for the maximum norm). We would like 1(i) to be small, 1(i) ≤ 1max , where 1max is a user-defined threshold value, so that the surrogate model is a sufficiently good representation of Rf . It is normally feasible to build a surrogate model satisfying the above requirements. As Rc is physically based, its response is similar to that of Rf . An appropriate surrogate model can be then created by introducing a sufficient number of preassigned parameters, for example, dielectric constants and substrate heights corresponding, if necessary, to individual components of the microwave structure in question, or even synthetic parameters (i.e. parameters not corresponding to any physical parameter in Rf but used to increase model flexibility, for example, a small capacitor introduced between coupled lines in the microstrip filter model). A rule of thumb is that the number of parameters should be larger than the number of design variables. If necessary, other space mapping transformations (e.g. input, frequency) can be incorporated (cf. Section 2.2). 3.3
Adaptively constrained parameter extraction
According to the algorithm proposed in [33], the PE process (3) is replaced by the following constrained version x(i) p = arg
min Rf (x(i) ) − Rc (x(i) , xp )
l (i) ≤xp ≤u(i)
(7)
where l(i)and u(i)are lower and upper bounds for the preassigned parameters at iteration i. We assume here that l(i) ¼ x(i21) – d(i)and u(i) ¼ x(i21) + d(i), where x(i21) is the p p p vector of model parameters at iteration i– 1 (x(0) p represents the initial values of the preassigned parameters), whereas d(i) is a vector representing the parameter space size (d(0) is a user-defined initial value). At iteration i, our algorithm adjusts l(i)and u(i) and performs PE as follows (d (i), x(i21) and 1max are input arguments): p 1. 2. 3. 4. 5. 6.
2 d(i) and u(i) ¼ x(i21) + d(i); Calculate l(i) ¼ x(i21) p p (i) Find xp using (7); If 1(i) ≤ adecr.1max then d(i+1) ¼ d(i)/bdecr; go to 6; If 1(i) . aincr.1max then d(i+1) ¼ d(i).bincr; go to 6; Set d(i+1) ¼ d(i); END.
Here, adecr , aincr , bdecr and bincr are user-defined parameters (typical values: adecr ¼ 1, aincr ¼ 2, bdecr ¼ 5, bincr ¼ 2). Our algorithm tightens the PE constraints if the approximation error is sufficiently small; loosens them otherwise. Note that constraint tightening improves the generalisation capability of the surrogate model: low approximation error 1(i21) ¼ Rf(x(i21))–Rc(x(i21), x(i21) )p and 1(i) ¼ Rf(x(i)) p (i) (i) –Rc(x , xp )p (satisfied when determining x(i21) and x(i) p p , IET Microw. Antennas Propag., 2011, Vol. 5, Iss. 10, pp. 1156–1163 doi: 10.1049/iet-map.2010.0607
www.ietdl.org respectively) makes it more likely to have Rf(x(i21)) – (i) Rc(x(i21), x(i) is reduced (because small p )p small if d (i) (i) (i21) xp – xp )1 ≤ d 1 implies the similarity of subsequent surrogate models). 3.4
Constrained surrogate optimisation
The convergence properties of the space mapping algorithm can be explicitly controlled by constraining the surrogate optimisation (2), which can be formulated as x(i+1) = arg
min
x, x−x(i) ≤d(i)
U (Rc (x, x(i) p ))
(8)
with a , 1 (recommended where d ¼ ax – x values are a ¼ 0.60.9; values too small could result in premature convergence without finding a satisfactory design). Note that (8) is mostly used as a safeguard because good convergence should be ensured by the algorithm of Section 3.3 for constrained PE. On the other hand, constrained surrogate optimisation is useful in finding a new design in the close neighbourhood of the current design x(i) (an over-flexible surrogate model may result in many designs that are good with respect to specification error, but are not necessarily close to x(i)). (i)
3.5
(i)
4
Verification examples
(i21)
Relation to trust region approach
It should be emphasised that neither the procedure of Section 3.4 nor (8) are related to the trust region approach [35] that has thus far been used to safeguard convergence for space mapping algorithms [32]. The latter reduces the search range for the surrogate model. It rejects a new design if it does not bring sufficient improvement with respect to the fine model specification [36]. This increases the computational cost of space mapping optimisation. According to the algorithm proposed here, the new design is never rejected: the generalisation of the surrogate model is accommodated by the adaptive PE procedure. 3.6
the convergence properties of the space mapping algorithm in two ways: (i) by reducing the value of Mi in x(i+1) – p (i+1) x(i) – x(i), which is enforced by the value p ≤ Mix of d(i) which, under normal circumstances, is being reduced from iteration to iteration, and (ii) by explicitly controlling x(i+1) – x(i) by the constrained surrogate optimisation process (8), which ensures that x(i+1) – x(i) , x(i) – x(i21). Note that the reduction of Mi is realised without degrading the approximation capability of the surrogate model because the constrained PE algorithm ensures that Rf(x(i)) – Rc(x(i), x(i) p )p is sufficiently small.
Convergence properties
Based on Theorem 1, Corollary 1 and the discussion in Section 3.1, the proposed technique is expected to improve
In this section we provide a comprehensive numerical verification of the algorithm for constrained space mapping described in Section 3. Also, we compare it with the standard space mapping algorithm [5] as well as with space mapping enhanced by the trust region approach [32]. We are particularly interested in the computational cost of the optimisation process, convergence properties of the algorithm, as well as the quality of the final design found. 4.1
Coupled microstrip bandpass filter
Consider the coupled microstrip bandpass filter [37] shown in Fig. 1a. The design parameters are x ¼ [L1 L2 L3 L4 S1 S2]T mm. The fine model is simulated in FEKO [38]. The coarse model, Fig. 1b, is an equivalent circuit implemented in Agilent ADS [39]. The design specifications are |S21| ≥ – 3 dB for 2.3 GHz ≤ v ≤ 2.5 GHz, and |S21| ≤ – 20 dB for 1.8 GHz ≤ v ≤ 2.15 GHz and 2.65 GHz ≤ v ≤ 3.0 GHz. The initial design is x(0) ¼ [24 4 14 12 0.2 0.1]T mm. The space mapping surrogate model has 14 preassigned parameters: 1r1 ¼ 1r. Clin1 ¼ 1r. Clin3 , 1r2 ¼ 1r.TL4 ¼ 1r.TL9 , 1r3 ¼ 1r.Clin2 ¼ 1r.Clin4 , 1r4 ¼ 1r.TL7 , H1 ¼ HClin1 ¼ HClin3 , H2 ¼ HTL4 ¼ HTL9 , H3 ¼ HClin2 ¼ HClin4 , H4 ¼ HTL7 , and dL1 , dL2 , dL3 , dL4 , dS1 , dS2 (design variable perturbations). Thus, we have xp ¼ [1r1 1r2 1r3 1r4 H1 H2 H3 H4 dL1 dL2 dL3 dL4 dS1 dS2]T. Symbols 1r.Elem and HElem refer to the dielectric constant (initial value 3.0) and substrate height (initial value 0.51 mm) of element Elem, respectively. Initial values for other parameters are zero. We take
Fig. 1 Coupled microstrip bandpass filter a Geometry [37] b Coarse model (Agilent ADS) IET Microw. Antennas Propag., 2011, Vol. 5, Iss. 10, pp. 1156– 1163 doi: 10.1049/iet-map.2010.0607
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Fig. 2 Coupled microstrip bandpass filter a Responses of the fine model (solid line) and the coarse model (dashed line) at the initial design b Responses of the fine model (solid line) and the space mapping surrogate model (circles) at the initial design after PE Note the excellent match between the models
d(0) ¼ [1 1 1 1 0.3 0.3 0.3 0.3 0.4 0.4 0.4 0.4 0.02 0.02]T (in respective units). Fig. 2a shows the fine and coarse model responses at the initial design. Thanks to the large number of surrogate model parameters, we obtain an excellent match to the fine model as illustrated in Fig. 2b. Fig. 3 shows the fine model response at the final design obtained using the proposed algorithm. Optimisation results for the standard and trust-regionenhanced space mapping as well as for the proposed constrained algorithm of Section 3 are summarised in Table 1 as well as in Fig. 4. We observe that the standard algorithm exhibits features typical of space mapping: fast initial progress, then stagnation, as indicated by the lack of or slow convergence, and oscillation in the specification error values. On the other hand, the proposed algorithm exhibits both a nice convergence pattern and consistent behaviour with respect to specification error. The trustregion convergence improves the convergence properties of
Fig. 3 Coupled microstrip bandpass filter: fine model response at the final design obtained using the proposed algorithm Table 1
Coupled bandpass filter: optimisation results
Algorithm
standard space mapping trust-region space mapping this work (constrained algorithm)
Specification error, dB Best found
Final
– 0.9 – 2.1 – 2.1
–0.4 –2.1 –2.1
Number of fine model evaluations
21a 21b 12
a
Algorithm terminated after 20 iterations without convergence Terminated after 21 fine model evaluations (good convergence pattern but tolerance requirements not fulfilled yet)
b
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the algorithm, but at the expense of extra computational effort: any designs that do not reduce the specification error are rejected, and so typically more than one fine model evaluation per iteration is required for this algorithm. The total optimisation cost is almost twice as high as for the constrained algorithm. 4.2
Wideband ring resonator bandpass filter
The second example is the wideband ring resonator bandpass filter [40] shown in Fig. 5a. The design parameters are x ¼ [L1 L2 L3 W1 W2 S]T mm. The fine model is simulated in FEKO [38]. The coarse model (Fig. 5b) is implemented in Agilent ADS [39]. The design specifications are |S21| ≥ –1 dB for 3.0 GHz ≤ v ≤ 5.5 GHz, and |S21| ≤ –20 dB for 2.0 GHz ≤ v ≤ 2.7 GHz and 5.8 GHz ≤ v ≤ 6.5 GHz. The initial design is the coarse model optimal solution x(0) ¼ [7 6 5 0.5 0.1 0.2]T mm. The space mapping surrogate model has 14 preassigned parameters: 1r1 ¼ 1r.Clin1 ¼ 1r.Clin2 , 1r2 ¼ 1r.TL2 ¼ 1r.TL6 , 1r3 ¼ 1r.TL3 ¼ 1r.TL7 , 1r4 ¼ 1r.TL4 ¼ 1r.TL5 , H1 ¼ HClin1 ¼ HClin2 , H2 ¼ HTL2 ¼ HTL6 , H3 ¼ HTL3 ¼ HTL7 , H4 ¼ HTL4 ¼ HTL5 , and dL1 , dL2 , dL3 , dW1 , dW2 , dS (design variable perturbations). Initial values of dielectric constants and substrate heights are 10.8 and 0.635 mm, respectively. Initial values for the other parameters are zero. Fig. 6a shows the fine model responses at the initial design and the design found by the new algorithm. Results are summarised in Table 2 and in Fig. 6b. For this example, both the standard and the trust-region-enhanced algorithms fail to find a design satisfying the specifications. The constrained algorithm yields a good design in just six fine model evaluations. 4.3 Coupled half-wavelength stepped impedance resonator bandpass filter Our last example is the coupled half-wavelength stepped impedance resonator (SIR) bandpass filter [41] (Fig. 7a). The design parameters are x ¼ [L1 L2 L3 L4 S W1 W2]T mm. The fine model is simulated in FEKO [38]. The coarse model (Fig. 7b) is implemented in Agilent ADS [39]. The design specifications are |S21| ≥ –3 dB for 2.3 GHz ≤ v ≤ 2.5 GHz, and |S21| ≤ –20 dB for 1.5 GHz ≤ v ≤ 2.1 GHz and 2.7 GHz ≤ v ≤ 3.5 GHz. The initial design is x(0) ¼ [5 1 13 4 0.5 1 1]T mm. The surrogate model has eight preassigned parameters: 1r1 ¼ 1r.TL1 ¼ 1r.Taper1 ¼ 1r.TL6 ¼ 1r.Taper2 , 1r2 ¼ 1r.TL2 ¼ 1r.TL5 , 1r3 ¼ 1r.TL3 ¼ 1r.TL4 , 1r4 ¼ 1r.Clin1 ¼ 1r.Clin2 , H1 ¼ IET Microw. Antennas Propag., 2011, Vol. 5, Iss. 10, pp. 1156–1163 doi: 10.1049/iet-map.2010.0607
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Fig. 4 Coupled microstrip bandpass filter a Convergence plot for the standard space mapping (o), trust-region-enhanced space mapping (×) and for the proposed constrained algorithm (∗ ) b Specification error against iteration index for the standard space mapping (o) and for the proposed algorithm (∗ ) Note that there is more than one fine model evaluation per iteration for the trust-region algorithm (21 evaluations in total)
Fig. 5 Wideband ring resonator bandpass filter a Geometry [40] b Coarse model (Agilent ADS)
Fig. 6 Wideband ring resonator bandpass filter a Responses of the fine model at the initial design (dashed line) and the final design obtained using the proposed algorithm (solid line) b Convergence plot for the standard space mapping (o), trust-region-enhanced space mapping (×) and for the proposed constrained algorithm (∗ ) Note that there is more than one fine model evaluation per iteration for the trust-region algorithm (13 evaluations in total)
HClin1 ¼ HClin3 , H2 ¼ HTL4 ¼ HTL9 , H3 ¼ HClin2 ¼ HClin4 , H4 ¼ HTL7 . Initial values of dielectric constants and substrate heights are 4.4 and 0.8 mm, respectively. The fine model responses at the initial design and the design found by the algorithm for constrained space mapping are shown in Fig. 8a. Results are summarised in Table 3 and in Fig. 8b. In this case, all algorithms found designs of comparable quality. Nevertheless, owing to its good convergence properties, the constrained algorithm exhibits the lowest computational cost. IET Microw. Antennas Propag., 2011, Vol. 5, Iss. 10, pp. 1156– 1163 doi: 10.1049/iet-map.2010.0607
4.4
Discussion
The extensive numerical verification of the presented space mapping algorithm with constrained PE allows us to formulate the following remarks: † The convergence properties of our algorithm are substantially improved with respect to the standard space mapping as well as with respect to the trust-region-enhanced algorithm. 1161
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www.ietdl.org Table 2
Wideband ring resonator filter: optimisation results
Algorithm
standard space mapping trust-region space mapping this work (constrained algorithm)
Specification error, dB Best found
Final
+5.0 +5.0
+5.0 +5.0
– 0.45
–0.45
Number of fine model evaluations
21 13
Table 3
Coupled half-wavelength SIR bandpass filter: optimisation results Algorithm
a
6
standard space mapping trust-region space mapping this work (constrained algorithm)
Specification error, dB
Number of fine model evaluations
Best found
Final
–2.1 –2.0
– 1.9 – 2.0
21a 21b
–2.1
– 2.1
9
a
Algorithm terminated after 20 iterations without convergence a
† The quality of the final design produced by the proposed algorithm is comparable to or better than for the other algorithms considered. † The proposed algorithm offers substantial reduction in computational cost of the optimisation process compared with the standard and trust-region-enhanced algorithms. It should be emphasised that although iteration-wise, convergence of both trust-region-enhanced space mapping and the algorithm with constrained PE is similar (cf. Figs. 6, 11 and 15), the latter uses an entirely different mechanism to improve the quality of the design from iteration to iteration. In particular, the proposed algorithm controls generalisation properties of the surrogate model by adjusting the parameter space of the model (see Section 3),
Algorithm terminated after 20 iterations without convergence Terminated after 21 fine model evaluations (good convergence pattern but tolerance requirements not yet fulfilled
b
whereas the trust-region approach is based on rejecting designs that bring no improvement in terms of satisfying the design specifications. The first mechanism does not require any extra fine model evaluations. This results in reduced computational cost of the optimisation process. On the other hand, one needs to remember that the critical component of our algorithm is an over-flexible surrogate model which is able to match the fine model with sufficient accuracy. As indicated in Section 3.2, such a model can be realised by using a sufficient number of parameters. The balance between approximation and generalisation capabilities of the model is then taken care of by our algorithm.
Fig. 7 Coupled half-wavelength SIR bandpass filter a Geometry [41] b Coarse model (Agilent ADS)
Fig. 8 Coupled half-wavelength SIR bandpass filter a Responses of the fine model at the initial design (dashed line) and the final design obtained using the proposed algorithm (solid line) b Convergence plot for the standard space mapping (o), trust-region enhanced space mapping (×) and for the proposed constrained algorithm (∗ ) Note that there is more than one fine model evaluation per iteration for the trust-region algorithm (21 evaluations in total; convergence not obtained) 1162
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Conclusion
A robust space-mapping-based algorithm exploiting an adaptively constrained PE and surrogate optimisation procedure is presented. Our technique effectively alleviates the problem of selecting the surrogate model parameters, and improves convergence properties and overall performance of the space mapping optimisation process. An extensive numerical study performed in this paper also indicates that the proposed technique reduces the computational cost of the optimisation process both with respect to the standard implementation and the trust-regionenhanced space mapping algorithm.
6
Acknowledgments
The authors thank Agilent Technologies, Santa Rosa, CA, for making ADS available. This work was supported in part by the Icelandic Centre for Research (RANNIS) grant 110034021, and by the Natural Sciences and Engineering Research Council of Canada under Grants RGPIN7239-06 and STPGP381153-09, and by Bandler Corporation.
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